
One limitation of this work is that we focused on the case where least \((a,b)\) -majorized elements exist for each choice of \(a\)  and \(b\) . Moreover, the definition of these elements requires them to be optimal for any choice of separable convex objective structure. For most applications of interest, these conditions are unnecessarily strong. For instance, \(a\)  and \(b\)  usually represent quantities that are restricted to a specific set of values, e.g., due to physical limitations of the considered system. Moreover, often only a few objective structures are relevant and these arise directly from the structure and properties of the system. Thus, it is interesting to investigate whether the existence of least \((a,b)\) -majorized elements for specific (sets of) values of \(a\)  and \(b\)  can be characterized. In particular, an interesting question is whether such a characterization exists for \(b=0\) , i.e., when we only consider scaled objective functions. Analogously, it would be interesting to see whether limiting the definition of least \((a,b)\) -majorized elements to particular subsets of objective structures leads to a characterization of feasible sets that is broader than those considered in this article. We do expect, however, that answering these questions requires a framework quite different from \((a,b)\) -majorization since in the latter the focus is primarily on the (Schur-)convexity of the objective functions and to a significant lesser extent on specific functions and parameters.

Another limitation is that we focused on the case where the feasible sets are either compact and convex or bounded and hole-free, i.e., contain exactly all integral points in its convex hull. An interesting direction for future research is to investigate whether our results can be extended to feasible sets that do not have such a convexity property. Promising candidates for such sets and corresponding optimization problems include extensions of base polyhedra and resource allocation problems, e.g., jump systems [1]} and semi-continuous knapsack polytopes [2]}, [3]}. Another promising class of problems consists of those whose structure and optimal solutions depend primarily on the convexity of the objective function rather than explicitly on the function itself, e.g., discrete speed scaling problems [4]}.

Our results indicate a natural connection between \((a,b)\) -majorization and base polyhedra, between weak \((a,b)\) -submajorization and supermodular polyhedra, between weak \((a,b)\) -supermajorization and submodular polyhedra, and between weak absolute \((a,b)\) -majorization and bisubmodular polyhedra. A natural follow-up question is whether more of such pairs exists, i.e., whether we can define new special cases of \((a,b)\) -majorization that have similar connections to other, perhaps more general, polyhedra. A first class of suitable candidates for sets in such a pair are polyhedra that are in some sense obtained from submodular or base polyhdra, examples of which not already discussed in this article are polybasic polyhedra [5]}, skew-bisubmodular polyhedra [6]}, and \(k\) -submodular polyhedra [7]}. The reason for this is that essentially all our characterization results depend on the existence of least \((a,b)\) -majorized elements for base polyhedra.
Another way to identify promising candidate sets are those for which a greedy algorithm in the style of Edmonds' [8]} is optimal for linear optimization. This can be seen by considering the role of linear optimization in the conditional gradient method for solving convex optimization problems with continuously differentiable objective functions (see also [9]}, [10]}). It can be shown that, given vectors \(a \in \mathbb {R}^n_{>0}\)  and \(b \in \mathbb {R}^n\)  and an objective function of the form \(\sum _{i \in N} a_i \phi \left(\frac{x_i}{a_i} + b_i x_i \right)\) , there exists a sequence of iterate solutions that is a valid possible outcome of the classical Frank-Wolfe algorithm [11]} for any choice of continuously differentiable convex function \(\phi \) . Via an analogous analysis of this algorithm and the existence of a greedy algorithm for linear optimization over bisubmodular polyhedra [12]}, [13]}, one can show that there exists a vector that minimizes \(\sum _{i \in N} a_i \phi \left(\frac{x_i}{a_i} + b_i x_i \right)\)  over a given bisubmodular polyhedron for any choice of continuously differentiable even convex function \(\phi \) . We expect that similar analyses can be done for polyhedra for which similar greedy algorithms are optimal for linear optimization such as (generalized) skew-bisubmodular polyhedra [6]} and polyhedra generated by Monge and greedy matrices [15]}, [16]}.

